Quantum Algorithms for Multiscale Partial Differential Equations

Partial differential equation (PDE) models with multiple temporal/spatial scales are prevalent in several disciplines such as physics, engineering, and many others. These models are of great practical importance but notoriously difficult to solve due to prohibitively small mesh and time step sizes limited by the scaling parameter and CFL condition. Another challenge in scientific computing could come from curse-of-dimensionality. In this paper, we aim to provide a quantum algorithm, based on either direct approximations of the original PDEs or their homogenized models, for prototypical multiscale problems in partial differential equations (PDEs), including elliptic, parabolic and hyperbolic PDEs. To achieve this, we will lift these problems to higher dimensions and leverage the recently developed Schr\"{o}dingerization based quantum simulation algorithms to efficiently reduce the computational cost of the resulting high-dimensional and multiscale problems. We will examine the error contributions arising from discretization, homogenization, and relaxation, analyze and compare the complexities of these algorithms in order to identify the best algorithms in terms of complexities for different equations in different regimes.